Phase Transitions in Random Discrete Structures Mihyun Kang Institute of Optimization and Discrete Mathematics Graz University of Technology
Phase Transition in Computer Science Random k -SAT problem To decide whether or not a random k-CNF formula F k ( n , m ) with n variables and m clauses is satisfiable. Phase transition from satisfiability to unsatisfiability of F k ( n , m ) : m n ∼ 2 k ln 2 Computational time required to find a satisfying truth assignment increases drastically: n ∼ 2 k m k Mihyun Kang Phase Transitions in Random Discrete Structures
Phase Transition in Statistical Physics Ising model (mathematical model of ferromagnetism) (up or down) Spins are arranged in lattice which interact with nearest neighbours Mihyun Kang Phase Transitions in Random Discrete Structures
Phase Transition in Statistical Physics Ising model (mathematical model of ferromagnetism) (up or down) Spins are arranged in lattice which interact with nearest neighbours Ordered phase at low temperatures Disordered phase at high temperatures Mihyun Kang Phase Transitions in Random Discrete Structures
Percolation in Geography, Materials Science and Physics the passage of fluid or gas going through porous or disordered media Mihyun Kang Phase Transitions in Random Discrete Structures
Mathematical Models of Percolation Bond percolation: each bond (or edge) is either open with prob. p or closed with prob. 1 − p , independently Site percolation: each site (or vertex) is either occupied with prob. p or empty with prob. 1 − p , independently p < p c p > p c Bond Percolation on Square Lattice Site Percolation on Hexagonal Lattice Mihyun Kang Phase Transitions in Random Discrete Structures
Percolation on Complete Graph K n Binomial random graph G ( n , p ) each edge of the complete graph K n is open with probability p , independently of each other cf. G ( n , m ) : a graph sampled uniformly at random among all graphs on n vertices and m edges Paul Erd˝ os (1913 – 1996) Alfréd Rényi (1921 – 1970) Mihyun Kang Phase Transitions in Random Discrete Structures
Phase Transition in Random Graphs I. Binomial Random Graph G ( n , p ) ⊲ Galton-Watson Tree II. Random Planar Graphs ⊲ Internal Structure – Kernel III. Random Graph Processes ⊲ Differential Equations Method Mihyun Kang Phase Transitions in Random Discrete Structures
I. Binomial Random Graph G ( n , p ) Cycle threshold � p ≪ 1 0 if n P [ G ( n , p ) contains a cycle ] → p ≫ 1 1 if n complete p=1 p=log n /n connected p cycles p=1/n p=0 empty Threshold Evolution of G ( n , p ) Mihyun Kang Phase Transitions in Random Discrete Structures
Phase Transition of Largest Component Binomial random graph G ( n , p ) [ E RD ˝ OS –R ÉNYI 60 ] Let p = t / n for a constant t > 0. If t < 1, with probability tending to 1 as n → ∞ (whp) all the components have O ( log n ) vertices. If t > 1, whp there is a unique largest component of order Θ( n ) , while every other component has O ( log n ) vertices. ⊲ Component exposure: Breath-First Search & Galton-Watson Tree [ K ARP 90 ] Mihyun Kang Phase Transitions in Random Discrete Structures
Galton-Watson Tree Branching Process The number of children is given by i.i.d. random variable ∼ Po ( t ) . If t < 1, the process dies out with probability 1. If t > 1, with positive probability ρ the process continues forever. „small” component in G ( n , p ) „giant” component of size ρ n + o ( n ) „small” component in G ( n , p ) in G ( n , p ) where 1 − ρ = e − t ρ Mihyun Kang Phase Transitions in Random Discrete Structures
Extinction Probability of Galton-Watson process Let T be the total number of nodes created in the process. Suppose t > 1. Consider the probability generating function � i < ∞ P ( T = i ) z i . q ( z ) := It satisfies e − t t k P ( Po ( t ) = k ) q ( z ) k = z k ! q ( z ) k = ze t ( q ( z ) − 1 ) . � � q ( z ) = z k k k i < ∞ P ( T = i ) satisfies q ( 1 ) = e t ( q ( 1 ) − 1 ) . The extinction probability q ( 1 ) = � Since q ( 1 ) = 1 − ρ we have 1 − ρ = e − t ρ . Mihyun Kang Phase Transitions in Random Discrete Structures
Largest Component in G ( n , p ) Let L ( n ) be the number of vertices in the largest component in G ( n , p ) . [ P ITTEL –W ORMALD 05; B EHRISCH –C OJA -O GHLAN –K. 09; B OLLOBAS AND R IORDAN 12+] Local Limit Theorem Let p = t / n with t > 1. Then σ 2 := V ( L ( n )) = ( ρ ( 1 − ρ ) / ( 1 − t ( 1 − ρ )) 2 ) n . and E ( L ( n )) = ρ n For any integer k with k = ρ n + x where x = O ( √ n ) = O ( σ ) − x 2 1 � � exp P ( L ( n ) = k ) ∼ √ . 2 σ 2 2 π σ t √ n t √ n ⌊ ρ n − t √ n ⌋ ⌊ ρ n + t √ n ⌋ ⌊ ρ n ⌋ Mihyun Kang Phase Transitions in Random Discrete Structures
Critical Phase How big is the largest component in G ( n , p ) , when pn = 1 + ε for ε = o ( 1 ) ? Béla Bollobás Tomasz Łuczak Mihyun Kang Phase Transitions in Random Discrete Structures
Critical Phase How big is the largest component in G ( n , p ) , when pn = 1 + ε for ε = o ( 1 ) ? [ B OLLOBÁS 84; Ł UCZAK 90; J ANSON –K NUTH –Ł UCZAK –P ITTEL 93; B OLLOBÁS –R IORDAN 13+] If ε n 1 / 3 → −∞ , whp L ( n ) = o ( n 2 / 3 ) . If ε n 1 / 3 → λ , a constant, whp L ( n ) = Θ( n 2 / 3 ) . If ε n 1 / 3 → ∞ , whp L ( n ) = ( 1 + o ( 1 )) 2 ε n . n 2/3 n 2/3 2/3 n << ~ >> s n − 2 / 3 = ε n 1 / 3 ⊲ Uniform random graph G ( n , m ) : m = n / 2 + s , Mihyun Kang Phase Transitions in Random Discrete Structures
II. Random Planar Graphs Planar graphs A planar graph is a graph that can be embedded in the plane (without crossing edges). non−planar K 3,3 K 5 Random planar graphs [ F RIEZE 87; M C D IARMID –S TEGER –W ELSH 05 ] Let P ( n , m ) be a uniform random planar graph with n vertices and m edges. Mihyun Kang Phase Transitions in Random Discrete Structures
Phase Transition in Random Planar Graphs Let L ( n ) denote the number of vertices in the largest component in P ( n , m ) . Two critical periods [ K.– Ł UCZAK 12 ] Let m = n / 2 + s . If s n − 2 / 3 → −∞ , whp L ( n ) ≪ n 2 / 3 . If s n − 2 / 3 → ∞ , whp L ( n ) = ( 2 + o ( 1 )) s ≫ n 2 / 3 . Let m = n + r . If r n − 3 / 5 → −∞ , whp n − L ( n ) ≫ n 3 / 5 . If r n − 3 / 5 → ∞ , whp n − L ( n ) = Θ( n 3 / 2 r − 3 / 2 ) ≪ n 3 / 5 . 3/5 L(n) n−L(n) n ~ 2 s << Mihyun Kang Phase Transitions in Random Discrete Structures
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